Lemma 48.27.5. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $d$. The module $\omega _ X$ of Lemma 48.27.1 has the following properties

$\omega _ X$ is a dualizing module on $X$ (Section 48.22),

$\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[d]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts and distinguished triangles for $K \in D_\mathit{QCoh}(X)$,

there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^{d - i}(\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$.

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